BRST-Invariant Deformations of Geometric Structures in Sigma Models

Abstract

We study a Lie algebra of formal vector fields $W_n$ with its application to the perturbative deformed holomorphic symplectic structure in the A-model, and a Calabi-Yau manifold with boundaries in the B-model. We show that equivalent classes of deformations are describing by a Hochschild cohomology theory of the DG-algebra $\mathfrak{A}=(A,Q)$, $Q=\bar{\partial }+{\partial }_{\mathrm{deform}}$, which is defined to be the cohomology of $(-1{)}^{n}Q+d_{\mathrm{Hoch}}$. Here $\bar{\partial }$ is the initial non-deformed BRST operator while ${\partial }_{\mathrm{deform}}$ is the deformed part whose algebra is a Lie algebra of linear vector fields ${\mathrm{gl}}_n$. We show that equivalent classes of deformations are described by a Hochschild cohomology of $\mathfrak{A}$, an important geometric invariant of the (anti)holomorphic structure on $X$. We discuss the identification of the harmonic structure $(H{T}^{\bullet }(X);H{\Omega }_{\bullet }(X))$ of affine space $X$ and the group ${\rm Ext}_{X^{2}}^n({\cO}_{\triangle}, {\cO}_{\triangle})$ (the HKR isomorphism), and bulk-boundary deformation pairing.

13 pages, no figures

References in corpus (2)

• Three-dimensional topological field theory and symplectic algebraic geometry I
• Lectures on D-branes and Sheaves